Lamina comprising cube corner elements and retroreflective sheeting

ABSTRACT

The present invention is directed to lamina(e) comprising cube corner elements, a tool comprising an assembly of laminae and replicas thereof. The invention further relates to retroreflective sheeting.

RELATED APPLICATIONS

This application is a divisional application of Ser. No. 10/404,265,filed Apr. 1, 2003 now U.S. Pat. No. 7,152,983, now allowed, whichclaims priority to provisional U.S. Patent Application Ser. No.60/452,464 filed Mar. 6, 2003.

FIELD OF THE INVENTION

The present invention is directed to a lamina comprising cube cornerelements, a tool comprising an assembly of laminae and replicationsthereof including in particular retroreflective sheeting.

BACKGROUND OF THE INVENTION

Retroreflective materials are characterized by the ability to redirectlight incident on the material back toward the originating light source.This property has led to the widespread use of retroreflective sheetingfor a variety of traffic and personal safety uses. Retroreflectivesheeting is commonly employed in a variety of articles, for example,road signs, barricades, license plates, pavement markers and markingtape, as well as retroreflective tapes for vehicles and clothing.

Two known types of retroreflective sheeting are microsphere-basedsheeting and cube corner sheeting. Microsphere-based sheeting, sometimesreferred to as “beaded” sheeting, employs a multitude of microspherestypically at least partially embedded in a binder layer and havingassociated specular or diffuse reflecting materials (e.g., pigmentparticles, metal flakes or vapor coats, etc.) to retroreflect incidentlight. Due to the symmetrical geometry of beaded retroreflectors,microsphere based sheeting exhibits the same total light returnregardless of orientation, i.e. when rotated about an axis normal to thesurface of the sheeting. Thus, such microsphere-based sheeting has arelatively low sensitivity to the orientation at which the sheeting isplaced on a surface. In general, however, such sheeting has a lowerretroreflective efficiency than cube corner sheeting.

Cube corner retroreflective sheeting typically comprises a thintransparent layer having a substantially planar front surface and a rearstructured surface comprising a plurality of geometric structures, someor all of which include three reflective faces configured as a cubecorner element.

Cube corner retroreflective sheeting is commonly produced by firstmanufacturing a master mold that has a structured surface, suchstructured surface corresponding either to the desired cube cornerelement geometry in the finished sheeting or to a negative (inverted)copy thereof, depending upon whether the finished sheeting is to havecube corner pyramids or cube corner cavities (or both). The mold is thenreplicated using any suitable technique such as conventional nickelelectroforming to produce tooling for forming cube cornerretroreflective sheeting by processes such as embossing, extruding, orcast-and-curing. U.S. Pat. No. 5,156,863 (Pricone et al.) provides anillustrative overview of a process for forming tooling used in themanufacture of cube corner retroreflective sheeting. Known methods formanufacturing the master mold include pin-bundling techniques, directmachining techniques, and techniques that employ laminae.

In pin bundling techniques, a plurality of pins, each having a geometricshape such as a cube corner element on one end, are assembled togetherto form a master mold. U.S. Pat. No. 1,591,572 (Stimson) and U.S. Pat.No. 3,926,402 (Heenan) provide illustrative examples. Pin bundlingoffers the ability to manufacture a wide variety of cube cornergeometries in a single mold, because each pin is individually machined.However, such techniques are impractical for making small cube cornerelements (e.g. those having a cube height less than about 1 millimeter)because of the large number of pins and the diminishing size thereofrequired to be precisely machined and then arranged in a bundle to formthe mold.

In direct machining techniques, a series of grooves are formed in thesurface of a planar substrate (e.g. metal plate) to form a master moldcomprising truncated cube corner elements. In one well known technique,three sets of parallel grooves intersect each other at 60 degreeincluded angles to form an array of cube corner elements, each having anequilateral base triangle (see U.S. Pat. No. 3,712,706 (Stamm)). Inanother technique, two sets of grooves intersect each other at an anglegreater than 60 degrees and a third set of grooves intersects each ofthe other two sets at an angle less than 60 degrees to form an array ofcanted cube corner element matched pairs (see U.S. Pat. No. 4,588,258(Hoopman)). In direct machining, a large number of individual faces aretypically formed along the same groove formed by continuous motion of acutting tool. Thus, such individual faces maintain their alignmentthroughout the mold fabrication procedure. For this reason, directmachining techniques offer the ability to accurately machine very smallcube corner elements. A drawback to direct machining techniques,however, has been reduced design flexibility in the types of cube cornergeometries that can be produced, which in turn affects the total lightreturn.

In techniques that employ laminae, a plurality of thin sheets (i.e.plates) referred to as laminae having geometric shapes formed on onelongitudinal edge, are assembled to form a master mold. Techniques thatemploy laminae are generally less labor intensive than pin bundlingtechniques because fewer parts are separately machined. For example, onelamina can typically have about 400-1000 individual cube cornerelements, in comparison to each pin having only a single cube cornerelement. However, techniques employing laminae have less designflexibility in comparison to that achievable by pin bundling.Illustrative examples of techniques that employ laminae can be found inEP 0 844 056 A1 (Mimura et al.); U.S. Pat. No. 6,015,214 (Heenan etal.); U.S. Pat. No. 5,981,032 (Smith); and U.S. Pat. No. 6,257,860(Luttrell).

The base edges of adjacent cube corner elements of truncated cube cornerarrays are typically coplanar. Other cube corner element structures,described as “full cubes” or “preferred geometry (PG) cube cornerelements”, typically comprise at least two non-dihedral edges that arenot coplanar. Such structures typically exhibit a higher total lightreturn in comparison to truncated cube corner elements. Certain PG cubecorner elements may be fabricated via direct machining of a sequence ofsubstrates, as described in WO 00/60385. However, it is difficult tomaintain geometric accuracy with this multi-step fabrication process.Design constraints may also be evident in the resulting PG cube cornerelements and/or arrangement of elements. By contrast, pin bundling andtechniques that employ laminae allow for the formation of a variety ofshapes and arrangements of PG cube corner elements. Unlike pin bundling,however, techniques that employ laminae also advantageously provide theability to form relatively smaller PG cube corner elements.

The symmetry axis of a cube corner is a vector that trisects thestructure, forming an equal angle with all three cube faces. In theaforementioned truncated cubes of Stamm, the symmetry axis is normal tothe equilateral base triangle and the cubes are considered to have nocant or tilt. The nomenclature “forward canting” or “positive canting”has been used in the cube corner arts to describe truncated cube cornerelements canted in a manner that increases only one base triangleincluded angle relative to 60°. Conversely, the nomenclature “backwardcanting” or “negative canting” has been used in the cube corner arts todescribe cube corner elements canted in a manner that increases two ofthe included angles of the base triangle relative to 60°. See U.S. Pat.No. 5,565,151 (Nilsen) and U.S. Pat. No. 4,588,258 (Hoopman). Canting ofPG cube corner elements is described in U.S. Pat. No. 6,015,214 (Heenanet al.).

Canting cube corner elements either backward or forward enhancesentrance angularity. Full cube corner elements have a higher total lightreturn than truncated cube corner elements for a given amount of cant,but the full cubes lose total light return more rapidly at higherentrance angles. One benefit of full cube corner elements is highertotal light return at low entrance angles, without substantial loss inperformance at higher entrance angles.

A common method for improving the uniformity of total light return (TLR)with respect to orientation is tiling, i.e. placing a multiplicity ofsmall tooling sections in more than one orientation in the finalproduction, as described for example in U.S. Pat. No. 4,243,618 (VanArnam), U.S. Pat. No. 4,202,600; and U.S. Pat. No. 5,936,770 (Nestegardet al.). Tiling can be visually objectionable. Further, tiling increasesthe number of manufacturing steps in making the tooling employed formanufacture of the sheeting.

In addition to being concerned with the TLR, the performance ofretroreflective sheeting also relates to the observation angularity ordivergence profile of the sheeting. This pertains to the spread of theretroreflected light relative to the source, i.e. typically, vehicleheadlights. The spread of retroreflected light from cube corners isdominated by effects including diffraction, polarization, andnon-orthogonality. For this purpose, it is common to introduce angleerrors such as described in Table 1 of column 5 of U.S. Pat. No.5,138,488 (Szczech).

Similarly, Example 1 of EP 0 844 056 A1 (Mimura) describes a fly cuttingprocess in which the bottom angles of V-shaped grooves formed with adiamond cutting tool were slightly varied in regular order, three typesof symmetrical V-shaped grooves having depths of 70.6 μm, 70.7 μm and70.9 μm were successively and repeatedly cut at a repeating pitch of141.4 μm in a direction perpendicular to the major surfaces of thesheets. Thus, a series of successive roof-shaped projections havingthree different vertical angles of 89.9°, 90.0°, and 91.0° in arepeating pattern were formed on one edge of the sheets.

Although the art describes a variety of retroreflective designs andtheir measured or calculated retroreflective performance; industry wouldfind advantage in retroreflective sheeting having new cube corneroptical designs and methods of manufacturing, particularly thosefeatures that contribute to improved performance and/or improvedmanufacturing efficiencies.

SUMMARY OF THE INVENTION

In one embodiment, the invention discloses a lamina comprising cubecorner elements having faces formed from grooves wherein adjacentgrooves range from being nominally parallel to nonparallel by less than1°. The adjacent grooves have included angles that differ by at least2°. In one aspect the included angles of the grooves are arranged in arepeating pattern. In another aspect, the faces of the elementsintersect at a common peak height. In yet another aspect, the grooveshave bisector planes that range from being mutually nominally parallelto nonparallel by less than 1°.

In another embodiment, the invention discloses a lamina comprisingpreferred geometry cube corner elements wherein at least a portion ofthe cube corner elements are canted having an alignment angle selectedfrom alignment angles between 45° and 135°, alignment angles between225° and 315°, and combinations thereof. Preferably, a first cube cornerelement is canted having an alignment angle between 60° and 120° and asecond adjacent cube is canted having an alignment angles between 240°and 300°. Further, the alignment angle of the first cube preferablydiffers from 0° or 180° by substantially the same amount as thealignment angle of the second cube differs.

In each of these embodiments, the cube corner elements preferablycomprise faces formed from alternating pairs of side grooves. Theincluded angle of each pair of side grooves preferably has a sum ofsubstantially 180°. Further, the included angle of a first groove ispreferably greater than 90° by an amount of at least about 5° (e.g.about 10° to about 20°) and the included angle of a second adjacentgroove is less than 90° by about the same amount.

In another embodiment, the invention discloses a lamina having amicrostructured surface comprising cube corner elements having facesformed from a side groove set wherein at least two grooves within theset are nonparallel by amounts ranging from greater than nominallyparallel to about 1°. The elements preferably comprise dihedral angleerrors having magnitudes between 1 arc minute and 60 arc minutes. Thedihedral angle errors are preferably arranged in a repeating pattern.The grooves comprise skew and/or inclination that vary in sign and ormagnitude.

In all disclosed embodiments, the adjacent grooves are preferably sidegrooves. Further, the elements preferably each have a face in a commonplane that defines a primary groove face. In addition, the elements arepreferred geometry cube corner elements.

In other embodiments, the invention discloses a master tool comprising aplurality of any one or combination of described lamina. The laminae arepreferably assembled such that cube corner elements of adjacent laminaeare in opposing orientations. The elements preferably have a shape inplan view selected from trapezoids, rectangles, parallelograms,pentagons, and hexagons.

In other embodiments, the invention discloses replicas of the mastertool including multigenerational tooling and retroreflective sheeting.The retroreflective sheeting may be derived from the laminae or have thesame optical features described with reference to a lamina.Retroreflective sheeting may have cube corner elements, cube cornercavities, or combinations thereof.

Hence, in other embodiments, the invention discloses retroreflectivesheeting comprising a row of preferred geometry cube corner elementshaving faces defined by grooves wherein adjacent side grooves range frombeing nominally parallel to nonparallel by less than 1° and haveincluded angles that differ by at least 2°. In other embodiments, theretroreflective sheeting comprises a row of cube corner elements whereina first cube corner element is canted having an alignment angle between45° and 135° and a second adjacent cube is canted having an alignmentangles between 225° and 315°. In yet other embodiments, theretroreflective sheeting comprises a row of preferred geometry cubecorner elements having faces defined by a side groove set wherein atleast two grooves within the set are nonparallel by amounts ranging fromgreater than nominally parallel to about 1°. In each of theseembodiments, the sheeting preferably further comprises the featuresdescribed with reference to the lamina or laminae.

In another aspect, the invention discloses retroreflective sheetingcomprising a pair of adjacent rows of preferred geometry cube cornerelements wherein adjacent elements in a row have at least one dihedraledge that ranges from being nominally parallel to nonparallel by lessthan 1° and wherein the pair of rows comprise at least two types ofmatched pairs.

In preferred embodiments, the retroreflective sheeting disclosed hasimproved properties. In one embodiment, the retroreflective sheetingexhibits a uniformity index of at least 1. Such uniformity can beobtained without tiling in more than one orientation. The uniformityindex is preferably at least 3 and more preferably at least 5. In otherpreferred embodiments, the retroreflective sheeting comprises an arrayof preferred geometry cube corner elements that exhibits an averagebrightness at 0° and 90° orientation according to ASTM D4596-1a of atleast 375 candelas/lux/m² for an entrance angle of −4° and anobservation angle of 0.5°. Preferably, the sheeting exhibits improvedbrightness at other observation angles as well.

The invention further discloses any combination of features describedherein.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of an exemplary single lamina prior toformation of cube corner elements.

FIG. 2 is an end view of an exemplary single lamina following theformation of a first groove set.

FIG. 3 is a side view of an exemplary single lamina following theformation of a first groove set.

FIG. 4 is a top view of an exemplary single lamina following theformation of a first groove set and a second groove set.

FIG. 5 is a top view of an exemplary single lamina following theformation of a first groove set and primary groove face.

FIG. 6 is a top plan view of an exemplary assembly of four laminaecomprising a first groove set and a third primary groove wherein thecube corners have been canted sideways.

FIG. 7 is a side view depicting the symmetry axes of a pair of adjacentsideways canted cubes on a lamina.

FIG. 8 is a perspective view of four laminae wherein the cube cornershave been canted sideways.

FIG. 9 is a perspective view of four laminae wherein the cube cornershave been canted sideways and the laminae have been assembled inopposing orientations.

FIG. 10 a is a representation of a backward canted cube corner element.

FIG. 10 b is a representation of a sideways canted cube corner element.

FIG. 10 c is a representation of a forward canted cube corner element.

FIG. 11 depicts a top plan view of an assembly of laminae wherein thecube corners have been canted forward in a plane normal to the edge ofthe lamina.

FIG. 12 depicts a top plan view of an assembly of laminae wherein thecube corners have been canted backward in a plane normal to the edge ofthe lamina.

FIG. 13 depicts an isointensity plot showing the predicted light returncontours for a matched pair of cube corner elements comprised ofpolycarbonate that have been canted forward 9.74°.

FIG. 14 depicts an isointensity plot showing the predicted light returncontours for a matched pair of cube corner elements comprised ofpolycarbonate that have been canted backward 7.74°.

FIG. 15 depicts an isointensity plot showing the predicted light returncontours for two opposing laminae that comprise polycarbonate cubes thathave been canted sideways 4.41°.

FIG. 16 depicts an isointensity plot showing the predicted light returncontours for two opposing laminae that comprise polycarbonate cubes thathave been canted sideways 5.23°.

FIG. 17 depicts an isointensity plot showing the predicted light returncontours for two opposing laminae that comprises polycarbonate cubesthat have been canted sideways 6.03°.

FIG. 18 depicts an isointensity plot showing the predicted light returncontours for two opposing laminae that comprise polycarbonate cubes thathave been canted sideways 7.33°.

FIG. 19 depicts an isointensity plot showing the predicted light returncontours for an assembly of laminae that comprises polycarbonate cubesthat have been canted sideways 9.74°.

FIG. 20 is a plot of alignment angle versus uniformity index.

FIG. 21 depicts a top plan view of a lamina having skewed side grooves.

FIG. 22 depicts each of the dihedral angles of a representative cubecorner element.

FIG. 23 depicts a side view of a cube corner element of a laminadepicting positive and negative inclination.

FIG. 24 depicts a spot diagram for cubes that are backward canted by7.47 degrees with angle errors of the primary groove ranging from 2 to10 arc minutes.

FIG. 25 depicts a spot diagram for cubes that are backward canted by7.47 degrees with angle errors of the side grooves ranging from 0 to −20arc minutes.

FIG. 26 depicts a spot diagram for cubes that are backward canted by7.47 degrees with a combination of primary groove and side groove angleerrors.

FIG. 27 depicts a spot diagram for cubes that are backward canted by7.47 degrees wherein the side grooves comprise a constant skew of 7 arcminutes, a side groove angle error of +1.5 arc minutes and inclinationvaried in a repeating pattern over every four grooves.

FIG. 28 depicts a spot diagram for cubes of the same geometry as FIG. 29except that the skew is −7 arc minutes rather than +7 arc minutes.

FIG. 29 depicts a spot diagram for the combination of FIG. 27 and FIG.28.

FIG. 30 comprises the same angle errors, skews, and inclinations as FIG.29 except that the cubes are forward canted by 7.47 degrees.

FIG. 31 depicts a spot diagram for cubes that are sideways canted by6.02 degrees having various skews and inclinations.

The drawings, particularly of the lamina(e), are illustrative and thusnot necessary representative of actual size. For example the drawing(s)may be an enlarged lamina or enlarged portion of a lamina.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention relates to a lamina and laminae comprising cubecorner elements, a tool comprising an assembly of laminae and replicas.There invention further relates to retroreflective sheeting.

The retroreflective sheeting is preferably prepared from a master moldmanufactured with a technique that employs laminae. Accordingly, atleast a portion and preferably substantially all the cube cornerelements of the lamina(e) and retroreflective sheeting are full cubesthat are not truncated. In one aspect, the base of full cube elements inplan view are not triangular. In another aspect, the non-dihedral edgesof full cube elements are characteristically not all in the same plane(i.e. not coplanar). Such cube corner elements are preferably “preferredgeometry (PG) cube corner elements”.

A PG cube corner element may be defined in the context of a structuredsurface of cube corner elements that extends along a reference plane.For the purposes of this application, a PG cube corner element means acube corner element that has at least one non-dihedral edge that: (1) isnonparallel to the reference plane; and (2) is substantially parallel toan adjacent non-dihedral edge of a neighboring cube corner element. Acube corner element whose three reflective faces comprise rectangles(inclusive of squares), trapezoids or pentagons are examples of PG cubecorner elements. “Reference plane” with respect to the definition of aPG cube corner element refers to a plane or other surface thatapproximates a plane in the vicinity of a group of adjacent cube cornerelements or other geometric structures, the cube corner elements orgeometric structures being disposed along the plane. In the case of asingle lamina, the group of adjacent cube corner elements consists of asingle row or pair of rows. In the case of assembled laminae, the groupof adjacent cube corner elements includes the cube corner elements of asingle lamina and the adjacent contacting laminae. In the case ofsheeting, the group of adjacent cube corner elements generally covers anarea that is discernible to the human eye (e.g. preferably at least 1mm²) and preferably the entire dimensions of the sheeting.

“Entrance angle” refers to the angle between the reference axis (i.e.the normal vector to the retroreflective sample) and the axis of theincident light.

“Orientation” refers to the angle through which the sample may berotated about the reference axis from the initial zero degreeorientation of a datum mark.

Lamina(e) refers to at least two lamina. “Lamina” refers to a thin platehaving length and height at least about 10 times its thickness(preferably at least 100, 200, 300, 400, 500 times its thickness). Theinvention is not limited to any particular dimensions of lamina(e). Inthe case of lamina intended for use in the manufacture ofretroreflective sheeting, optimal dimensions may be constrained by theoptical requirements of the final design (e.g. cube corner structures).In general the lamina has a thickness of less than 0.25 inches (6.35 mm)and preferably less than 0.125 inches (3.175 mm). The thickness of thelamina is preferably less than about 0.020 inches (0.508 mm) and morepreferably less than about 0.010 inches (0.254 mm). Typically, thethickness of the lamina is at least about 0.001 inches (0.0254 mm) andmore preferably at least about 0.003 inches (0.0762 mm). The laminaranges in length from about 1 inch (25.4 mm) to about 20 inches (50.8cm) and is typically less than 6 inches (15.24 cm). The height of thelamina typically ranges from about 0.5 inches (12.7 mm) to about 3inches (7.62 cm) and is more typically less than about 2 inches (5.08cm).

With reference to FIGS. 1-8 lamina 10 includes a first major surface 12and an opposing second major surface 14. Lamina 10 further includes aworking surface 16 and an opposing bottom surface 18 extending betweenfirst major surface 12 and second major surface 14. Lamina 10 furtherincludes a first end surface 20 and an opposing second end surface 22.In a preferred embodiment, lamina 10 is a right rectangular polyhedronwherein opposing surfaces are substantially parallel. However, it willbe appreciated that opposing surfaces of lamina 10 need not be parallel.

Lamina 10 can be characterized in three-dimensional space bysuperimposing a Cartesian coordinate system onto its structure. A firstreference plane 24 is centered between major surfaces 12 and 14. Firstreference plane 24, referred to as the x-z plane, has the y-axis as itsnormal vector. A second reference plane 26, referred to as the x-yplane, extends substantially coplanar with working surface 16 of lamina10 and has the z-axis as its normal vector. A third reference plane 28,referred to as the y-z plane, is centered between first end surface 20and second end surface 22 and has the x-axis as its normal vector. Forthe sake of clarity, various geometric attributes of the presentinvention will be described with reference to the Cartesian referenceplanes as set forth herein. However, it will be appreciated that suchgeometric attributes can be described using other coordinate systems orwith reference to the structure of the lamina.

The lamina(e) of the present invention preferably comprise cube cornerelements having faces formed from, and thus comprise, a first grooveset, an optional second groove set, and preferably a third primarygroove (e.g. primary groove face).

FIGS. 2-9 illustrate a structured surface comprising a plurality of cubecorner elements in the working surface 16 of lamina 10. In general, afirst groove set comprising at least two and preferably a plurality ofgrooves 30 a, 30 b, 30 c, etc. (collectively referred to as 30) areformed in working surface 16 of lamina 10. The grooves 30 are formedsuch that the respective groove vertices 33 and the respective firstreference edges 36 extend along an axis that intersects the first majorsurface 12 and the working surface 16 of lamina 10. Although workingsurface 16 of the lamina 10 may include a portion that remains unaltered(i.e. unstructured), it is preferred that working surface 16 issubstantially free of unstructured surface portions.

The direction of a particular groove is defined by a vector aligned withthe groove vertex. The groove direction vector may be defined by itscomponents in the x, y and z directions, the x-axis being perpendicularto reference plane 28 and the y-axis being perpendicular to referenceplane 24. For example, the groove direction for groove 30 b is definedby a vector aligned with groove vertex 33 b. It is important to notethat groove vertices may appear parallel to each other in top plan vieweven though the grooves are not parallel (i.e. different z-directioncomponent).

As used herein, the term “groove set” refers to grooves formed inworking surface 16 of the lamina 10 that range from being nominallyparallel to non-parallel to within 1° to the adjacent grooves in thegroove set. As used herein “adjacent groove” refers to the closestgroove that is nominally parallel or non-parallel to within 1°.Alternatively or in addition thereto, the grooves of a groove set mayrange from being nominally parallel to non-parallel to within 1° toparticular reference planes as will subsequently be described.Accordingly, each characteristic with regard to an individual grooveand/or the grooves of a groove set (e.g. perpendicular, angle, etc.)will be understood to have this same degree of potential deviation.Nominally parallel grooves are grooves wherein no purposeful variationhas been introduced within the degree of precision of the groove-formingmachine. The grooves of the groove set may also comprise smallpurposeful variations for the purpose of introducing multiplenon-orthogonality (MNO) such as included angle errors, and/or skew,and/or inclination as will subsequently be described in greater detail.

Referring to FIGS. 3-9, the first groove set comprises grooves 30 a, 30b, 30 c, etc. (collectively referred to by the reference numeral 30)that define first groove surfaces 32 a, 32 b, 32 c, etc. (collectivelyreferred to as 32) and second groove surfaces 34 b, 34 c, 34 d, etc.(collectively referred to as 34) that intersect at groove vertices 33 b,33 c, 33 d, etc. (collectively referred to as 33). At the edge of thelamina, the groove forming operation may form a single groove surface 32a.

In another embodiment depicted in FIG. 4, lamina 10 may optionallyfurther comprise a second groove set comprising at least two andpreferably a plurality of adjacent grooves, collectively referred to as38) also formed in the working surface 16 of lamina 10. In thisembodiment, the first and second groove sets intersect approximatelyalong a first reference plane 24 to form a structured surface includinga plurality of alternating peaks and v-shaped valleys. Alternatively,the peaks and v-shaped valleys can be off-set with respect to eachother. Grooves 38 define third groove surfaces 40 a, 40 b, etc.(collectively referred to as 40) and fourth groove surfaces 42 b, 42 c,etc. (collectively referred to as 42) that intersect at groove vertices41 b, 41 c, etc. (collectively referred to as 41) as shown. At the edgeof the lamina, the groove forming operation may form a single groovesurface 40 a.

Both these first and second groove sets may also be referred to hereinas “side grooves”. As used herein side grooves refer to a groove setwherein the groove(s) range from being nominally parallel tonon-parallel to with 1°, per their respective direction vectors, to theadjacent side grooves of the side groove set. Alternatively or inaddition thereto, side grooves refers to a groove that range from beingnominally parallel to reference plane 28 to nonparallel to referenceplane 28 to within 1°. Side grooves are typically perpendicular toreference plane 24 to this same degree of deviation in plan view.Depending on whether the side grooves are nominally parallel ornon-parallel within 1°, individual elements in the replicated assembledmaster typically have the shape of trapezoids, rectangles,parallelograms and pentagons, and hexagons when viewed in plan view witha microscope or by measuring the dihedral angles or parallelism of theside grooves with an interferometer. Suitable interferometers willsubsequently be described.

Although the third face of the elements may comprise working surface 12or 14, such as describe in EP 0 844 056 A1 (Mimura et al.), the laminapreferably comprises a primary groove face 50 that extends substantiallythe full length of the lamina. Regardless of whether the third face is aworking surface (i.e. 12 or 14) of the lamina or a primary groove face,the third face of each element within a row preferably share a commonplane. With reference to FIGS. 5-6 and 8-9, primary groove face 50ranges from being nominally perpendicular to faces 32, 34, 40 and 42 tonon-perpendicular to with 1°. Formation of primary groove face 50results in a structured surface that includes a plurality of cube cornerelements having three perpendicular or approximately perpendicularoptical faces on the lamina. A single lamina may have a single primarygroove face, a pair of groove faces on opposing sides and/or a primarygroove along the intersection of working surface 16 with reference plane24 that concurrently provides a pair of primary groove faces (e.g. FIG.4). The primary groove is preferably parallel to reference plane 26 towithin 1°.

A pair of single laminae with opposing orientations and preferablymultiple laminae with opposing orientations are typically assembled intoa master tool such that their respective primary groove faces form aprimary groove. For example, as depicted in FIGS. 6 and 8-9, fourlaminae (i.e. laminae 100, 200, 300 and 400 are preferably assembledsuch that every other pair of laminae are positioned in opposingorientations (i.e. the cube corner elements of lamina 100 are inopposing orientation with the cube corner elements of lamina 200 and thecube corner elements of lamina 300 are in opposing orientation with thecube corner elements of lamina 400). Further, the pairs of laminaehaving opposing orientation are positioned such that their respectiveprimary groove faces 50 form primary groove 52. Preferably the opposinglaminae are positioned in a configuration (e.g. 34 b aligned with 42 b)in order to minimize the formation of vertical walls.

After formation of the groove sets, working surface 16 ismicrostructured. As used herein, “microstructured” refers to at leastone major surface of the sheeting comprising structures having a lateraldimension (e.g. distance between groove vertices of the cube cornerstructures) of less than 0.25 inches (6.35 mm), preferably less than0.125 inches (3.175 mm) and more preferably less than 0.04 inches (1mm). The lateral dimension of cube corner elements, is preferably lessthan 0.020 inches (0.508 mm) and more preferably less than 0.007 inches(0.1778 mm). Accordingly, the respective groove vertices 33 and 41 arepreferably separated by this same distance throughout the groove otherthan the small variations resulting from non-parallel grooves. Themicrostructures have an average height ranging from about 0.001 inches(0.0254 mm) to 0.010 inches (0.254 mm), with a height of less than 0.004inches (0.1016 mm) being most typical. Further, the lateral dimension ofa cube corner microstructure is typically at least 0.0005 inches (0.0127mm). Cube corner microstructures may comprise either cube cornercavities or, preferably, cube corner elements having peaks.

In one embodiment, as depicted in FIG. 3-9, the present inventionrelates to a lamina and laminae comprising a side groove set whereinadjacent grooves comprise different included angles. “Included angle”refers to the angle formed between the two faces of a V-shaped groovethat intersect at the groove vertex. The included angle is typically afunction of the geometry of the diamond-cutting tool and its positionrelative to the direction of cut. Hence, a different diamond tool istypically employed for each different included angle. Alternatively, yetmore time consuming, different included angles may be formed by makingmultiple cuts within the same groove. The included angles of a firstgroove (e.g. side groove) differs from an adjacent groove (e.g. secondside groove) by at least 2° (e.g. 3°, 4°, 5°, 6°, 7°, 8°, 9°) preferablyat least 10° (e.g. 11°, 12°, 13°, 14°), and more preferably by at least15° (e.g. 16°, 17°, 18°, 19°, 20°). Accordingly, the difference inincluded angle is substantially larger than differences in includedangles that would arise from purposeful angle errors introduced for thepurpose of non-orthogonality. Further, the difference in included anglesis typically less than 70° (e.g. 65°, 60°, 50°), preferably less than55°, more preferably less than 50°, and even more preferably less than40°.

In one aspect, the differing included angles (e.g. of adjacent sidegrooves) are arranged in a repeating pattern to minimize the number ofdifferent diamond cutting tools needed. In such embodiment, the sum ofadjacent side groove angles is about 180°. In a preferred embodiment,the lamina comprises a first sub-set of side grooves having an includedangle greater than 90° alternated with second sub-set of side grooveshaving an included angle less than 90°. In doing so, the included angleof a first groove is typically greater than 90° by an amount of at leastabout 5°, and preferably by an amount ranging from about 10° to about20°; whereas the included angle of the adjacent groove is less than 90°by about the same amount.

Although, the lamina may further comprise more than two sub-sets and/orside grooves having included angles of nominally 90°, the lamina ispreferably substantially free of side grooves having an included angleof nominally 90°. In a preferred embodiment, the lamina comprises analternating pair of side grooves (e.g. 75.226° and 104.774°) and thus,only necessitates the use of two different diamonds to form the totalityof side grooves. Accordingly, with reference to FIGS. 6-9, every otherside grooves, i.e. 30 a, 30 c, 30 e, etc. has an included angle of75.226° alternated with the remaining side grooves, i.e. 30 b, 30 d,etc. having an included angle of 104.774°. As will subsequently bedescribed in further detail, employing differing included angles in thismanner improves the uniformity index.

In another aspect, alternatively or in combination with the differingincluded angles (e.g. of adjacent side grooves) being arranged in arepeating pattern, the resulting cube corner elements have faces thatintersect at a common peak height, meaning that cube peaks (e.g. 36) arewithin the same plane to within 3-4 microns. It is surmised that acommon peak height contribute to improved durability when handling thetooling or sheeting by evenly distributing the load.

Alternatively or in combination thereof, the lamina comprises sidewayscanted cube corner elements. For cube corner elements that are solelycanted forward or backward, the symmetry axes are canted or tilted in acant plane parallel with reference plane 28. The cant plane for a cubecorner element is the plane that is both normal to reference plane 26and contains the symmetry axis of the cube. Accordingly, the normalvector defining the cant plane has a y component of substantially zerofor cube corner elements that are solely canted forward or backward. Inthe case of cube corner elements that are solely canted sideways, thesymmetry axes of the cubes are canted in a plane that is substantiallyparallel to reference plane 24 and thus, the normal vector defining thecant plane has an x component of substantially zero.

The projection of the symmetry axis in the x-y plane may alternativelybe used to characterize the direction of cant. The symmetry axis isdefined as the vector that trisects the three cube corner faces formingan equal angle with each of these three faces. FIGS. 10 a-10 c depictthree different cube corner geometries in plan view that are solelybackward canted, solely sideways canted, and solely forward canted,respectively. In these figures the cube peak extends out of the page andthe projection of the symmetry axis (extending into the page from thecube peak) in the x-y plane is shown by the arrow. The alignment angleis measured counterclockwise in this view from the dihedral edge 11(e.g. dihedral 2-3) that is approximately normal to a side of the cubein plan view. In the case of backward canting in the absence of sidewayscanting, the alignment angle is 0 degrees, whereas in the case offorward canting in the absence of sideways canting the alignment angleis 180 degrees. For a cube that has been canted sideways in the absenceof forward or backward canting, the alignment angle is either 90° (asshown in FIG. 10 b) or 270°. Alignment angle is 90° when the projectionof the symmetry axis points to the right (FIG. 10 b) and 270° when theprojection of the symmetry axis points to the left.

Alternatively, the cube may be canted such that the cant plane normalvector comprises both an x-component and y-component (i.e. x-componentand y-component are each not equal to zero). At an alignment anglebetween 0° and 45° or between 0° and 315° the backward cant component ispredominant with the backward cant component and sideways cant componentbeing equal at an alignment angle of 45° or 315°. Further at analignment angle between 135° and 225°, the forward cant component ispredominant with the forward cant component and sideways cant componentbeing equal at 135° and at 225°. Accordingly, cant planes comprising apredominant sideways cant component have an alignment angle between 45°and 135° or between 225° and 315°. Hence, a cube corner element ispredominantly sideways canting when the absolute value of they-component of the cant plane normal vector is greater than the absolutevalue of the x-component of the cant plane normal vector.

For embodiments wherein the sideways canted cubes are formed from analternating pair of side grooves having different included angle cubeswhere the cant plane is parallel to reference plane 24 the adjacentcubes within a given lamina (e.g. α-β or α′-β′) are canted in the sameor parallel planes. However, in general, if there is an x component tothe cant plane normal vector, then adjacent cubes within a particularlamina are not canted in the same plane. Rather, the cube corner matchedpairs are canted in the same or parallel planes (i.e. α-α′ or β-β′).Preferably, the cube corner elements of any given lamina have only twodifferent alignment angles, e.g. derived from adjacent side groovescomprising different included angles. The alignment angle for thesideways canting example in FIG. 10 b is 90°, corresponding to the β-β′cubes in FIG. 6. Similarly, the alignment angle for the α-α′ sidewayscanted cubes in FIG. 6 is 270° (not shown).

FIG. 11 depicts laminae wherein the cubes are canted forward; whereasFIG. 12 depicts laminae wherein the cubes are canted backward. Cubedesigns canted in this manner result in a single type of matchedopposing cube pairs. The cube 54 a of FIG. 11 with faces 64 a and 62 bis the same as the cube 54 b with faces 64 b and 62 c that is the sameas cube 54 c with faces 64 c and 62 d, etc. Accordingly, all the cubeswithin the same row are the same, the row being parallel to referenceplane 24. The cube 54 a is a matched opposing pair with cube 56 a havingfaces 66 e and 68 d. Further, the cube 54 b is a matched opposing pairwith cube 56 b. Likewise, cube 54 c is a matched opposing pair with cube56 c. Similarly, cube 57 of FIG. 12 is a matched opposing pair with cube58. Matched pairs result when 180° rotation of a first cube about anaxis normal to the plane of the structured surface will result in a cubethat is super-imposable onto a second cube. Matched pairs need notnecessarily be directly adjacent or contacting within the group of cubecorner elements. Matched pairs typically provide retroreflectiveperformance that is symmetric with respect to positive or negativechanges in entrance angle for opposing orientations.

In contrast, sideways canting results in a cube design comprising twodifferent cube orientations within the same row and thus created by thesame side groove set. For a single lamina comprising both the first andsecond set of side grooves or a pair of adjacent laminae assembled inopposing orientations, the laminae comprise four distinctly differentcubes and two different matched pairs, as depicted in FIGS. 6, 8-9. Thefour distinctly different cubes are identified as cubes alpha (α) havingfaces 32 b and 34 c, beta (β) having faces 32 c and 34 d, alpha prime(α′) having faces 40 c and 42 d, and beta prime (β′) having faces 40 band 42 c. Cubes (α, α′) are a matched pair with the same geometry whenrotated 180° such that the cube faces are parallel, as are cubes (β,β′). Further, the cubes on adjacent laminae (e.g. 100, 200) areconfigured in opposing orientations. Although the symmetry axis of thecubes is tipped sideways, the bisector plane of the side grooves (i.e.the plane that divides the groove into two equal parts) preferablyranges from being nominally parallel to the bisector plane of anadjacent side groove (i.e. mutually parallel) to being nonparallelwithin 1°. Further, each bisector plane ranges from being nominallyparallel to reference plane 28 to being nonparallel to reference plane28 within 1°.

FIGS. 13-14 are isobrightness contour graphs illustrating the predictedtotal light return for a retroreflective cube corner element matchedpair formed from a material having an index of refraction of 1.59 atvarying entrance angles and orientation angles. In FIG. 13 the matchedpair is forward canted 9.74° (e.g. cube corner elements 54, 56 of FIG.11). In FIG. 14, the matched pair is backward canted 7.47° (e.g. cubecorner elements 57, 58 of FIG. 12). FIGS. 15-19 are isobrightnesscontour graph illustrating the predicted total light return for laminaecomprising retroreflective cube corner elements formed from a materialhaving an index of refraction of 1.59 at varying entrance angles andorientation angles where the cube corner elements are canted sideways4.41°, 5.23°, 6.03°, 7.33°, and 9.74°, respectively for alignment anglesof 90° and 270°. An alternating pair of side grooves (i.e. 75.226° and104.774°) is utilized for FIG. 17 to produce cube corner elements thatare sideways canted by 6.03°. The sideways canted arrays have two typesof matched pairs, the β-β′ and α-α′ as depicted in FIG. 6. These matchedpairs have alignment angles of 90° and 270° respectively. In each ofFIGS. 15-19, the isobrightness contour graph is for laminae having thesame (i.e. vertical) orientation as depicted in FIGS. 6, 11 and 12.

Predicted total light return for a cube corner matched pair array may becalculated from a knowledge of percent active area and ray intensity.Total light return is defined as the product of percent active area andray intensity. Total light return for directly machined cube cornerarrays is described by Stamm U.S. Pat. No. 3,812,706.

For an initial unitary light ray intensity, losses may result from twopass transmissions through the front surface of the sheeting and fromreflection losses at each of the three cube surfaces. Front surfacetransmission losses for near normal incidence and a sheeting refractiveindex of about 1.59 are roughly 0.10 (roughly 0.90 transmission).Reflection losses for cubes that have been reflectively coated dependfor example on the type of coating and the angle of incidence relativeto the cube surface normal. Typical reflection coefficients for aluminumreflectively coated cube surfaces are roughly 0.85 to 0.9 at each of thecube surfaces. Reflection losses for cubes that rely on total internalreflection are essentially zero (essentially 100% reflection). However,if the angle of incidence of a light ray relative to the cube surfacenormal is less than the critical angle, then total internal reflectioncan break down and a significant amount of light may pass through thecube surface. Critical angle is a function of the refractive index ofthe cube material and of the index of the material behind the cube(typically air). Standard optics texts such as Hecht, “Optics”, 2ndedition, Addison Wesley, 1987 explain front surface transmission lossesand total internal reflection. Effective area for a single or individualcube corner element may be determined by, and is equal to, thetopological intersection of the projection of the three cube cornersurfaces on a plane normal to the refracted incident ray with theprojection of the image surfaces of the third reflection on the sameplane. One procedure for determining effective aperture is discussed forexample by Eckhardt, Applied Optics, v. 10, n. 7, July 1971, pg.1559-1566. Straubel U.S. Pat. No. 835,648 also discusses the concept ofeffective area or aperture. Percent active area for a single cube cornerelement is then defined as the effective area divided by the total areaof the projection of the cube corner surfaces. Percent active area maybe calculated using optical modeling techniques known to those ofordinary skill in the optical arts or may be determined numericallyusing conventional ray tracing techniques. Percent active area for acube corner matched pair array may be calculated by averaging thepercent active area of the two individual cube corner elements in thematched pair. Alternatively stated, percent active aperture equals thearea of a cube corner array that is retroreflecting light divided by thetotal area of the array. Percent active area is affected for example bycube geometry, refractive index, angle of incidence, and sheetingorientation.

Referring to FIG. 13 vector V₁ represents the plane that is normal toreference plane 26 and includes the symmetry axes of cube cornerelements 54, 56 in FIG. 11. For example, V₁ lies in a planesubstantially normal to the primary groove vertex 51 formed by theintersection of the primary groove faces 50. The concentricisobrightness curves represent the predicted total light return of thearray of cube corner elements 54, 56 at various combinations of entranceangles and orientation angles. Radial movement from the center of theplot represents increasing entrance angles, while circumferentialmovement represents changing the orientation of the cube corner elementwith respect to the light source. The innermost isobrightness curvedemarcates the set of entrance angles at which a matched pair of cubecorner elements 54, 56 exhibit 70% total light return. Successivelyoutlying isobrightness curves demarcate entrance and orientation angleswith successively lower percentages.

A single matched pair of forward or backward canted cubes typically havetwo planes (i.e. V₁ and V₂) of broad entrance angularity that aresubstantially perpendicular to one another. Forward canting results inthe principle planes of entrance angularity being horizontal andvertical as shown in FIG. 13. The amount of light returned at higherentrance angles varies considerably with orientation and is particularlylow between the planes of best entrance angularity. Similarly, backwardcanting results in the principle planes of entrance angularity (i.e. V₃and V₄) oriented at roughly 45° to the edge of the lamina as shown inFIG. 14. Similarly, the amount of light returned at higher entranceangles varies considerably with orientation and is particularly lowbetween the planes of best entrance angularity.

FIGS. 15-19 depict the predicted total light return (TLR) isointensitycontours for a pair of opposing laminae having sideways canted cubes.The light return is more uniform as indicated by the shape of the plotapproaching circular, in comparison to the isointensity plots of forwardand backward canted cubes of FIGS. 13 and 14. FIGS. 15-19 depictsubstantially higher light return at the locations of low light returnevident in FIGS. 13 and 14. Accordingly, good retention of TLR at higherentrance angles (up to at least 45° entrance) is predicted. Thisimproved orientation performance is desirable for signing products sincethe signs are commonly positioned at orientations of 0°, 45° and 90°.Prior to the present invention, a common method for improving theuniformity of total light return with respect to orientation has beentiling, i.e. placing a multiplicity of small tooling sections in morethan one orientation, such as described for example in U.S. Pat. No.5,936,770. Sideways canted cube corner arrays can improve the uniformityof total light return, without the need for tiling and thus the array issubstantially free of tiling in more than one orientation.

In order to compare the uniformity of total light return (TLR) ofvarious optical designs, the average TLR at orientations of 0°, 45° and90° may be divided by the range of TLR at orientations of 0°, 45° and90°, i.e. the difference between the maximum and minimum TLR at theseangles, all at a fixed entrance angle. The entrance angle is preferablyat least 30° or greater, and more preferably 40° or greater. Preferreddesigns exhibit the maximum ratio of average TLR relative to TLR range.This ratio, i.e. “uniformity index (UT)” was calculated for a 40°entrance angle for the forward and backward canted cubes of FIGS. 13 and14, respectively as well as for the sideways canted cubes having variousdegrees of tilt of FIGS. 15-19. For Table 1 the spacing of the sidegrooves is equal to the thickness of the lamina (i.e. aspect ratio=1).The calculated uniformity index is summarized in Table 1 as follows:

TABLE 1 Sideways Forward Backward (alignment angle = 90°) Amount of cant9.74 7.47 4.41 5.23 6.03 7.33 9.74 (arc minutes) Avg. TLR 0.210 0.1330.160 0.184 0.209 0.180 0.166 (0/45/90) TLR Range 0.294 0.154 0.0900.023 0.034 0.167 0.190 (0/45/90) UI 0.71 0.87 1.79 8.02 6.23 1.08 0.88

${{Uniformity}{\mspace{11mu}\;}{Index}\mspace{14mu}({UI})} = \frac{{{Average}\mspace{14mu}{TLR}\mspace{14mu}{of}\mspace{14mu} 0^{\circ}},45^{\circ},90^{\circ}}{{{Range}\mspace{14mu}{at}\mspace{14mu} 0^{\circ}},{45^{\circ}\mspace{14mu}{and}\mspace{14mu} 90^{\circ}}}$

Improved orientation uniformity results when the uniformity index isgreater than 1. Preferably, the uniformity index is greater than 3 (e.g.4), and more preferably greater than 5 (e.g. 6, 7, 8). Uniformity indexwill vary as a function of variables such as cube geometry (e.g. amountand type of cant, type of cube, cube shape in plan view, location ofcube peak within aperture, cube dimensions), entrance angle, andrefractive index.

FIG. 20 illustrates the uniformity index plotted versus alignment anglefor canted cube corner arrays with varying amounts of cant and varying xand y components for their cant plane normal vectors. The arrays havetwo types of matched pairs, similar to the β-β′ and α-α′ as depicted inFIG. 6, although these cubes and/or pairs may not be mutually adjacent.The cubes in each matched pair have substantially the same alignmentangle. Alignment angles for the two types of matched pairs differ from0° or 180° by the same amount. For example, for an alignment angle of100° (differing from 180° by 80°) for a first cube or first matched pairthe second (e.g. adjacent) cube or second matched pair would have analignment angle of 260° (also differing from 180° by 80°).

FIG. 20 shows that the average TLR for polycarbonate (having an index ofrefraction of 1.59) as well as the uniformity index are maximized forcube geometries having a predominant sideways canting component, i.e.the range roughly centered about alignment angles of 90° and 270°. Notethat alignment angles between 0° and 180° are presented on the X orhorizontal axis of FIG. 20 from left to right. Alignment anglesincreasing from 180° to 360° degrees are plotted from right to left.

Preferably, the alignment angle is greater than 50° (e.g. 51°, 52°, 53°,54°), more preferably greater than 55° (e.g. 56°, 57°, 58°, 59°), andeven more preferably greater than 60°. Further the alignment angle ispreferably less than 130° (e.g. 129°, 128°, 127°, 126°) and morepreferably less than 125° (e.g. 124°, 123°, 122°, 121°), and even morepreferably less than 120°. Likewise the alignment angle is preferablygreater than 230° (e.g. 231°, 232°, 233°, 234°), and more preferablygreater than 235° (e.g. 236°, 237°, 238°, 239°), and even morepreferably greater than 240°. Further the alignment angle is preferablyless than 310° (e.g. 309°, 308°, 307°, 306°) and more preferably lessthan 305° (e.g. 304°, 303°, 302°, 301°) and even more preferably lessthan 300°.

The amount of tilt of the cube symmetry axes relative to a vectorperpendicular to the plane of the cubes is at least 2° and preferablygreater than 3°. Further, the amount of tilt is preferably less than 9°.Accordingly, the most preferred amount of tilt ranges from about 3.5° toabout 8.5° including any interval having end points selected from 3.6°,3.7°, 3.8°, 3.9°, 4.0°, 4.1°, 4.2°, 4.3°, 4.4° and 4.5° combined withend points selected from 7.5°, 7.6°, 7.7°, 7.8°, 7.9°, 8.0°, 8.1°, 8.2°,8.3° and 8.4°. Cube geometries that may be employed to produce thesediffering amounts of sideways cant are summarized in Table 2. Thealignment angle may be 90° or 270° for each amount of cant.

TABLE 2 Amount of Side groove Side groove Side groove Side groove CantSub-set 1 Sub-set 2 Sub-set 1 Sub-set 2 (°) Half angle (°) ½ angle (°)Full angle (°) Full angle (°) 4.41 39.591 50.409 79.182 100.818 5.2338.591 51.409 77.182 102.818 6.03 37.613 52.387 75.226 104.774 7.3336.009 53.991 72.018 107.982 9.74 33.046 56.954 66.092 113.908

Although differing included angles alone or in combination with thepreviously described sideways canting provide improved brightnessuniformity in TLR with respect to changes in orientation angle over arange of entrance angles, it is also preferred to improve theobservation angularity or divergence profile of the sheeting. Thisinvolves improving the spread of the retroreflected light relative tothe source (typically, vehicle headlights). As previously describedretroreflected light from cube corners spreads due to effects such asdiffraction (controlled by cube size), polarization (important in cubeswhich have not been coated with a specular reflector), andnon-orthogonality (deviation of the cube corner dihedral angles from 90°by amounts less than 1°). Spread of light due to non-orthogonality isparticularly important in (e.g. PG) cubes produced using laminae sincerelatively thin laminae would be required to fabricate cubes where thespreading of the return light was dominated by diffraction. Such thinlaminae are particularly difficult to handle during fabrication.

Alternatively, or in addition to the features previously described, inanother embodiment the present invention relates to an individuallamina, a master tool comprising the assembled laminae, as well asreplicas thereof including retroreflective replicas, comprising sidegrooves wherein the side grooves comprise “skew” and/or “inclination”.Skew and/or inclination provides cubes with a variety of controlleddihedral angle errors or multiple non-orthogonality (MNO) and thusimproves the divergence profile of the finished product. As used herein“skew” refers to the deviation from parallel with reference to referenceplane 28.

FIG. 21 shows an exaggerated example in plan view of a single laminawith one row of cube corner elements comprising skewed grooves. Sidegrooves 80 a and 80 b are cut with positive skew, grooves 80 c and 80 ewithout skew, and groove 80 d with negative skew. The path of the sidegrooves 80 has been extended in FIG. 21 for clarity. Provided sidegrooves 80 a, 80 c, and 80 e have the same included angle (e.g. 75°,first groove sub-set), the same cutting tool or diamond can be used toform all of these grooves. Further, if the alternating grooves, namely80 b and 80 d have the same included angle (e.g. 105°, second groovesub-set) 80 b and 80 d can be cut with a second diamond. The primarygroove face 50 may also be cut with one of these diamonds if the primarygroove half angle is sufficiently close to the side groove half anglefor the first or second sub-sets. Optionally, one of the cutting toolsmay be rotated during cutting of the primary groove face in order toachieve the correct primary groove half angle. The primary groove faceis preferably aligned with the side of the lamina. Thus, the entirelamina can be cut incorporating MNO with the use of only two diamonds.Within each groove set skew can be easily varied during machining toproduce a range of dihedral angles. Cube corners in general have threedihedral angles attributed to the intersections of the three cube faces.The deviation of these angles from 90° is commonly termed the dihedralangle error and may be designated by dihedral 1-2, dihedral 1-3, anddihedral 2-3. In one naming convention, as depicted in FIG. 22, cubedihedral angle 1-3 is formed by the intersection of groove surface 82with primary groove face 50, cube dihedral 1-2 is formed by theintersection of groove surface 84 with primary groove face 50, and cubedihedral 2-3 is formed by the intersection of groove surface 84 withgroove surface 82. Alternatively, the same naming convention may beemployed wherein the third face is working surface 12 or 14 rather thana primary groove face. For a given groove, positive skew (80 a, 80 b)results in decreasing dihedral 1-3 and increasing dihedral 1-2 whilenegative skew results in increasing dihedral 1-3 and decreasing dihedral1-2.

For example, with reference to FIG. 21 four different cubes are formed.The first cube 86 a has groove surfaces (i.e. faces) 82 a and 84 b, thesecond cube 86 b groove surfaces 82 b and 84 c, the third cube 86 cgroove surfaces 82 c and 84 d, and the fourth cube 86 d has groovesurfaces 82 d and 84 e. The intersection of groove surfaces 82 a, 82 b,and 84 d with groove face 50 are less than 90°, whereas the intersectionof groove surfaces 84 b and 82 d with groove face 50 are greater than90°. The intersection of groove surfaces 82 c, 84 c, and 84 e withgroove face 50 are 90° since grooves 80 c and 80 e are without skew. Thepreceding discussion assumes that the inclination (as will subsequentlybe defined) is the same for all the side grooves in FIG. 21 and equalszero. The (e.g. PG) cube corner elements are trapezoids orparallelograms in plan view shape as a result of using skewed groovesduring machining.

Alternatively, or in addition to the features previously described, theside grooves may comprise positive or negative inclination.“Inclination” refers to the deviation in slope in reference plane 28 ofa particular side groove from the nominal orthogonal slope (i.e. theslope of the vector normal to the primary groove surface). The directionof a side groove is defined by a vector aligned with the vertex of saidgroove. Orthogonal slope is defined as the slope in which the vertex 90of a groove would be parallel to the normal vector of groove face 50(normal to groove face 50) for skew equal to zero. In one possiblenaming convention, positive inclination 92 results in decreasing bothdihedral 1-3 and dihedral 1-2 for a given side groove while negativeinclination 94 results in increasing both dihedral 1-3 and dihedral 1-2.

Combining skew and/or inclination during machining provides significantflexibility in varying the dihedral angle errors of the cube cornerelements on a given lamina. Such flexibility is independent of cant.Accordingly skew and/or inclination may be employed with uncanted cubes,forward canted cubes, backward canted cubes, as well as sideways cantedcubes. The use of skew and/or inclination provides a distinct advantageas it can be introduced during the machining of individual laminawithout changing the tool (e.g. diamond) used to cut the side grooves.This can significantly reduce machining time as it typically can takehours to correctly set angles after a tool change. Furthermore, dihedral1-2 and dihedral 1-3 may be varied in opposition using skew and/orinclination. “Varied in opposition” as used herein is defined asintentionally providing within a given cube corner on a lamina dihedral1-2 and 1-3 errors (differences from 90°) that differ in magnitudeand/or sign. The difference in magnitude is typically at least ¼ arcminutes, more preferably at least ½ arc minutes, and most preferably atleast 1 arc minutes. Hence the grooves are nonparallel by amount greaterthan nominally parallel. Further, the skew and/or inclination is suchthat the magnitude is no more than about 1° (i.e. 60 arc minutes).Further, the (e.g. side) grooves may comprise a variety of differentcomponents of skew and/or inclination along a single lamina.

Dihedral angle errors may also be varied by changing the half angles ofthe primary or side grooves during machining. Half angle for sidegrooves is defined as the acute angle formed by the groove face and aplane normal to reference plane 26 that contains the groove vertex. Halfangle for primary grooves or groove faces is defined as the acute angleformed by the groove face and reference plane 24. Changing the halfangle for the primary groove results in a change in slope of groove face50 via rotation about the x-axis. Changing the half angle for a sidegroove may be accomplished via either changing the included angle of thegroove (the angle formed by opposing groove faces e.g. 82 c and 84 c) orby rotating a groove about its vertex. For example, changing the angleof the primary groove face 50 will either increase or decrease all ofthe dihedral 1-2 and dihedral 1-3 errors along a given lamina. Thiscontrasts to changes in inclination where the dihedral 1-2 and dihedral1-3 errors can be varied differently in each groove along a givenlamina. Similarly, the half angle for the side grooves may vary,resulting in a corresponding change in dihedral 2-3. Note that for sidegrooves that are orthogonal or nearly orthogonal (within about 1°) tothe primary groove face, dihedral 1-2 and dihedral 1-3 are veryinsensitive to changes in side groove half angle. As a result, varyingthe half angles of the primary or side grooves during machining will notallow dihedral 1-2 and dihedral 1-3 to vary in opposition within a givencube corner. Varying the half angles of the primary or side groovesduring machining may be used in combination with skew and/or inclinationto provide the broadest possible control over cube corner dihedral angleerrors with a minimum number of tool changes. While the magnitude of anyone of half angle errors, skew, or inclination can ranges up to about1°, cumulatively for any given cube the resulting dihedral angle erroris no more than about 1°.

For simplicity during fabrication, skew and/or inclination arepreferably introduced such that the dihedral angle errors are arrangedin patterns. Preferably, the pattern comprises dihedral angle errors 1-2and 1-3 that are varied in opposition within a given cube corner.

Spot diagrams are one useful method based on geometric optics ofillustrating the spread in the retroreflected light resulting fromnon-orthogonality from a cube corner array. Cube corners are known tosplit the incoming light ray into up to six distinct return spotsassociated with the six possible sequences for a ray to reflect from thethree cube faces. The radial spread of the return spots from the sourcebeam as well as the circumferential position about the source beam maybe calculated once the three cube dihedral errors are defined (see e.g.Eckhardt, “Simple Model of Cube Corner Reflection”, Applied Optics, V10,N7, July 1971). Radial spread of the return spots is related toobservation angle while circumferential position of the return spots isrelated to presentation angle as further described in US Federal TestMethod Standard 370 (Mar. 1, 1977). A non-orthogonal cube corner can bedefined by the surface normal vectors of its three faces. Return spotpositions are determined by sequentially tracking a ray as it strikesand reflects from each of the three cubes faces. If the refractive indexof the cube material is greater than 1, then refraction in and out ofthe front surface cube must also be taken into account. Numerous authorshave described the equations related to front surface reflection andrefraction (e.g. Hecht and Zajac, “Optics”, 2^(nd) edition, AddisonWesley 1987). Note that spot diagrams are based on geometric optics andhence neglect diffraction. Accordingly, cube size and shape is notconsidered in spot diagrams.

The return spot diagram for five different cubes that are backwardcanted by 7.47 degrees (e.g. FIG. 12) with errors in the primary groovehalf angle of five consecutive grooves of +2, +4, +6, +8, and +10 arcminutes is depicted in FIG. 24. The half angle errors for the sidegrooves are zero (dihedral 2-3=0) in this example, as are skew andinclination. All the side groove included angles are 90°. The verticaland horizontal axes in FIG. 24 correspond to reference planes 28 and 24,respectively. Note that changes in the slope of the primary groove faceresult in return spots located primarily along the vertical andhorizontal axes.

The dihedral errors as a function of primary groove half angle errorsare presented in Table 3 for the same errors used to produce FIG. 24.Note that dihedral 1-2 and dihedral 1-3 have the same magnitude and signand thus, do not vary in opposition.

TABLE 3 Primary Groove Error Dihedral 1-2 Dihedral 2-3 Dihedral 1-3 (arcminutes) (arc minutes) (arc minutes) (arc minutes) 2 1.4 0.0 1.4 4 2.80.0 2.8 6 4.2 0.0 4.2 8 5.7 0.0 5.7 10 7.1 0.0 7.1

The return spot diagram for the same type of backward canted cubes withdihedral 2-3 errors of −20, −15, −10, −5, and 0 arc minutes is depictedin FIG. 25. The half angle errors for the primary groove are zero(dihedral 1-3=dihedral 1-2=0) in this example, as are skew andinclination. As stated previously, variations in the half angles for theside grooves may be used to produce the dihedral 2-3 errors. Thedihedral 2-3 errors result in return spots located primarily along thehorizontal axis.

FIG. 26 depicts a return spot diagram resulting from combining primarygroove half angle errors with variations in the half angles for the sidegrooves for the same type of backward canted cubes as described withreference to FIGS. 24-25. In this example, the primary groove half angleerror is 10 arc minutes while dihedral 2-3 error is 0, 2, 4, and 6 arcminutes respectively for four different adjacent cubes on the lamina. Aconstant included angle error of +3 arc minutes could be used to producethese side grooves, with the opposing half angle errors as shown inTable 4. The return spots are again located primarily along the verticaland horizontal axes, with some spreading in the horizontal plane due tothe nonzero values for dihedral 2-3. Overall the return spot diagram islocalized and non-uniform.

The dihedral errors as a function of primary groove half angle errorsare presented in Table 4 for the errors used to produce FIG. 26. Notethat dihedral 1-2 and dihedral 1-3 have the same magnitude and sign andhence do not vary in opposition (i.e. are substantially free of varyingin opposition). Note that a given cube corner is formed by two adjacentside grooves and preferably a primary groove surface. The upper sidegroove in FIG. 22 forms dihedral 1-3 while the lower side groove formsdihedral 1-2. The intersection of the upper and lower side grooves formsdihedral 2-3. Side groove included angle is the sum of the upper andlower half angle errors for a groove that forms adjacent cubes (e.g.with reference to Table 4 the total included angle is +3 arc minutes andresults from adding the upper half angle of a first cube with the lowerhalf angle of the adjacent cube).

TABLE 4 Dihedral Dihedral Lower Upper Half 1-2 2-3 Dihedral HalfAngleAngle Cube (arc (arc 1-3 Error Error No. minutes) minutes) (arc minutes)(arc minutes) (arc minutes) 1 7.1 4.0 7.1 3 1 2 7.1 6.0 7.1 2 4 3 7.12.0 7.1 −1 3 4 7.1 0.0 7.1 0 0

The preceding examples (i.e. FIGS. 24-26) were for backward canted cubeswith varying half angle errors. In an analogous manner, forward cantedcubes can be shown to have qualitatively similar return spot diagramss,i.e. substantially non-uniform with spots localized especially along thehorizontal and vertical axes. Dihedral 1-2 and dihedral 1-3 of forwardcanted cubes also will have the same magnitude and sign and thus aresubstantially free of varying in opposition. In consideration of theuses of cube corner retroreflective sheeting, it is apparent thatlocalized, non-uniform spot diagramss (e.g. FIGS. 24-26) are generallyundesirable. Sheeting may be placed on signs in a wide variety oforientations, both as the background color as well as in some cases ascut out letters. Furthermore, signs may typically be positioned on theright, on the left, or above the road. In the case of vehicles markedwith retroreflective sheeting for conspicuity, the position of thevehicle relative to the viewer is constantly changing. Both the left andright headlights of a vehicle illuminate a retroreflective target, andthe position of the driver is quite different with respect to theseheadlights (differing observation and presentation angles). Vehiclessuch as motorcycles, where the driver is directly above the headlight,are commonly used. Finally, all of the relevant angles defining theviewing geometry change with distance of the driver/observer to theretroreflective sheeting or target. All of these factors make it clearthat a relatively uniform spread of return spots is highly desirable inretroreflective sheeting. Because of the flexibility to easily introducea wide range of dihedral angle errors, including dihedral 1-2 anddihedral 1-3 that vary in opposition, skew and/or inclination may beutilized to provide a relatively uniform spot return diagram.

FIG. 27 presents a return spot diagram resulting from variations in onlyinclination on a single lamina with the same backward canted cubes usedin FIGS. 24-26. Half angle errors for the side grooves are +1.5 arcminutes on each side (dihedral 2-3 and side groove angle error of +3 arcminutes) and primary groove error is zero. Skew is constant in thisexample at +7 arc minutes. Inclination is varied in a repeating patternover every four grooves (i.e. two grooves +5 arc minutes, then twogrooves −1 arc minute). The spot pattern is much more uniformlydistributed both radially (observation) and circumferentially(presentation) in comparison with FIGS. 24-26.

The dihedral errors for this example of varying inclination arepresented in Table 5. The order of machining of the inclinations (arcminutes) is −1, +5, +5, −1 in a repeating pattern. For example withreference to cube no. 1, the first side groove has an inclination of −1and the second side groove has an inclination of +5. Note that dihedral1-2 and dihedral 1-3 vary in opposition with different magnitudes(absolute value of the dihedral angle errors are unequal) and signs.

TABLE 5 Inclination Dihedral 1-2 Dihedral 3-2 Dihedral 1-3 Cube No. (arcminutes) (arc minutes (arc minutes) (arc minutes) 1 −1, 5 5.1 3.0 −7.9 2 5, 5 0.8 3.0 −7.9 3  5, −1 0.8 3.0 −3.7 4 −1, −1 5.1 3.0 −3.7

FIG. 28 depicts the return spot diagram resulting from the same geometryas FIG. 27, except skew is −7 arc minutes instead of +7 arc minutes forall side grooves. The spot diagram is again uniformly distributed incomparison with FIGS. 24-26 as well as complementary to the spot diagramshown in FIG. 27. The dihedral errors for this example of varyinginclination are presented in Table 6. Note once again that dihedral 1-2and dihedral 1-3 vary in opposition, differing both in magnitude and/orsign. The change in sign of the skew has resulted in a switch in themagnitude and sign of dihedral 1-2 and 1-3 in comparison to Table 5.

TABLE 6 Inclination Dihedral 1-2 Dihedral 3-2 Dihedral 1-3 (arc minutes)(arc minutes) (arc minutes) (arc minutes) −1, 5 −3.7 3.0 0.8  5, 5 −7.93.0 0.8  5, −1 −7.9 3.0 5.1 −1, −1 −3.7 3.0 5.1

The positive and negative skews of the two preceding examples may becombined, providing the spot diagram of FIG. 29. This combination mightbe achieved by machining half of the lamina with +7 arc minutes of skewand the other half with −7 arc minutes of skew. Alternatively, thepositive and negative skew could be combined within each lamina,resulting in both skew and inclination being varied concurrently withina given lamina. In the latter case, a small number of other return spotswould result from the cubes positioned at the boundary of the positiveand negative skew sections. The spot diagram is particularly uniformlydistributed in comparison with FIGS. 24-26 as it results from thecombination of the spot diagrams in FIGS. 27 and 28. A combination ofthe dihedral errors as shown in Tables 5 and 6 are associated with thisspot diagram, with dihedral 1-2 and dihedral 1-3 differing in magnitudeand sign, varying in opposition.

FIG. 30 presents the same half angle errors, skews, and inclinations asFIG. 29 except for cubes that are forward canted by 7.47°. The spotdiagram is also uniformly distributed although slightly different thanthe backward canted spot diagram of FIG. 29. The dihedral errorsassociated with this spot diagram are summarized in Table 7, wheredihedral 1-2 and dihedral 1-3 again vary in opposition, including atleast one cube where dihedral 1-2 and dihedral 1-3 differ in magnitudeand/or sign.

TABLE 7 Inclination Skew Dihedral 1-2 Dihedral 3-2 Dihedral 1-3 (arcminutes) (arc minutes) (arc minutes) (arc minutes) (arc minutes) −1, 5 74.3 3.0 −7.2  5, 5 7 0.1 3.0 −7.2  5, −1 7 0.1 3.0 −2.9 −1, −1 7 4.3 3.0−2.9 −1, 5 −7 −2.9 3.0 0.1  5, 5 −7 −7.2 3.0 0.1  5, −1 −7 −7.2 3.0 4.3−1, −1 −7 −2.9 3.0 4.3

The same skew and inclination combinations may also be utilizedadvantageously in combination with sideways canted cube corners toprovide a uniformly distributed spot diagram. Sideways canted cubes, aspreviously discussed, comprise two different cube orientations withinthe same row. Preferably, care should be taken to apply the combinationsof skew and/or inclination equally to both types of cube in a given row(e.g. alpha (α) and beta (β)) in order to obtain uniform performance atvarious entrance and orientation angle combinations. The return spotdiagram for cubes that are sideways canted by 6.03° (FIG. 6, alignmentangle 90° or 270°) utilizing skew and inclination is shown in FIG. 31.The same combinations of +7 and −7 arc minutes of skew with −1 and 5 arcminutes of inclination were applied equally to both the alpha (α) andbeta (β) cubes. Half angle errors for the side grooves are +1.5 arcminutes on each side (dihedral 2-3 and side groove angle error of +3 arcminutes) and primary groove error is zero. The spot diagram is veryuniformly distributed in observation and presentation angle. Thedihedral errors associated with this spot diagram are summarized inTable 8, where dihedral 1-2 and dihedral 1-3 again vary in opposition,including at least one cube where dihedral 1-2 and dihedral 1-3 differin magnitude and/or sign.

TABLE 8 Dihedral Dihedral Dihedral Skew Inclination Inclination 1-2 3-21-3 Lower Upper (arc (arc (arc (arc (arc (arc Included Included minutes)minutes minutes) minutes) minutes) minutes) angle (°) angle (°) 7 −1 −14.3 3.0 −3.9 52.387 37.613 7 −1 5 5.1 3.0 −7.4 37.613 52.387 7 5 5 −0.53.0 −7.6 52.387 37.613 7 5 −1 1.5 3.0 −2.7 37.613 52.387 7 −1 5 4.3 3.0−7.6 52.387 37.613 7 5 5 1.5 3.0 −7.4 37.613 52.387 7 5 −1 −0.5 3.0 −3.952.387 37.613 7 −1 −1 5.1 3.0 −2.7 37.613 52.387 −7 −1 −1 −3.9 3.0 4.337.613 52.387 −7 −1 5 −2.7 3.0 1.5 52.387 37.613 −7 5 5 −7.6 3.0 −0.537.613 52.387 −7 5 −1 −7.4 3.0 5.1 52.387 37.613 −7 −1 5 −3.9 3.0 −0.537.613 52.387 −7 5 5 −7.4 3.0 1.5 52.387 37.613 −7 5 −1 −7.6 3.0 4.337.613 52.387 −7 −1 −1 −2.7 3.0 5.1 52.387 37.613

A characteristic of the exemplary cube corner elements of Tables 5-8 isthe formation of at least one and typically a plurality of PG cubecorner elements in a row having three dihedral angle errors wherein thedihedral angle errors are different from each other. Anothercharacteristic is that the dihedral angle errors, and thus the skewand/or inclination, is arranged in a repeating pattern throughout alamina or row of adjacent cube corner elements. Further the adjacentlamina or row is preferably optically identical except rotated 180°about the z-axis forming pairs of laminae or pairs of rows.

Methods of machining laminae and forming a master tool comprising aplurality of laminae is known, such as described in U.S. Pat. No.6,257,860 (Lutrell et al.). For embodiments wherein the side grooves aresubstantially free of skew and/or inclination, side grooves may beformed in a plurality of stacked laminae, such as described in U.S. Pat.No. 6,257,860 (Lutrell et al.) and U.S. Pat. No. 6,159,407 (Krinke etal.).

Accordingly, further described herein are methods of machining laminaeby providing a lamina or laminae and forming V-shaped grooves on workingsurface 16 of the lamina wherein the grooves are formed with any one orcombinations of the features previously described.

In general, the lamina(e) may comprise any substrate suitable forforming directly machined grooves on the edge. Suitable substratesmachine cleanly without burr formation, exhibit low ductility and lowgraininess and maintain dimensional accuracy after groove formation. Avariety of machinable plastics or metals may be utilized. Suitableplastics comprise thermoplastic or thermoset materials such as acrylicsor other materials. Machinable metals include aluminum, brass, copper,electroless nickel, and alloys thereof. Preferred metals includenon-ferrous metals. Suitable lamina material may be formed into sheetsby for example rolling, casting chemical deposition, electro-depositionor forging. Preferred machining materials are typically chosen tominimize wear of the cutting tool during formation of the grooves.

The diamond tools suitable for use are of high quality such as diamondtools that can be purchased from K&Y Diamond (Mooers, N.Y.) or ChardonTool (Chardon, Ohio). In particular, suitable diamond tools are scratchfree with 10 mils of the tip, as can be evaluated with a 2000× whitelight microscope. Typically, the tip of the diamond has a flat portionranging in size from about 0.00003 inches (0.000762 mm) to about 0.00005inches (0.001270 mm). Further, the surface finish of suitable diamondtools preferably have a roughness average of less than about 3 nm and apeak to valley roughness of less than about 10 nm. The surface finishcan be evaluated by forming a test cut in a machinable substrate andevaluating the test cut with a micro-interferometer, such as can bepurchased from Wyko (Tucson, Ariz.), a division of Veeco.

The V-shaped grooves are formed with a diamond-tooling machine that iscapable of forming each groove with fine precision. Moore Special ToolCompany, Bridgeport, Conn.; Precitech, Keene, N.H.; and Aerotech Inc.,Pittsburg, Pa., manufacture suitable machines for such purpose. Suchmachines typically include a laser interferometer-positioning device. Asuitable precision rotary table is commercially available from AA Gage(Sterling Heights, Mich.); whereas a suitable micro-interferometer iscommercially available from Zygo Corporation (Middlefield, Conn.) andWyko (Tucson, Ariz.) a division of Veeco. The precision (i.e. point topoint positioning) of the groove spacing and groove depth is preferablyat least as precise as +/−500 nm, more preferably at least as precise as+/−250 nm and most preferably at least as precise as +/−100 nm. Theprecision of the groove angle is at least as precise as +/−2 arc minutes(+/−0.033 degrees), more preferably at least as precise as +/−1 arcminute (+/−0.017 degrees), even more preferably at least at precise as+/−½ arc minute (+/−0.0083 degrees), and most preferably at least asprecise as +/−¼ arc minute (+/−0.0042 degrees) over the length of thecut (e.g. the thickness of the lamina). Further, the resolution (i.e.ability of groove forming machine to detect current axis position) istypically at least about 10% of the precision. Hence, for a precision of+/−100 nm, the resolution is at least +/−10 nm. Over short distances(e.g. about 10 adjacent parallel grooves), the precision isapproximately equal to the resolution. In order to consistently form aplurality of grooves of such fine accuracy over duration of time, thetemperature of the process is maintained within +/−0.1° C. andpreferably within +/−0.01° C.

While the change in shape of a single cube corner element due to skewand/or inclination is small with respect to a single element (e.g.limited primarily to changes in the dihedral angles), it is evident thatforming skewed and/or inclined grooves in a stack of laminae may beproblematic. Since the side grooves deviate from parallel up to as muchas 1°, significantly varying cube geometries may be produced across thestack. These variations increase as the stack size increases. Thecalculated maximum number of laminae that can be machined concurrently(i.e. in a stack) without creating significantly varying cube geometriesis as few as two laminae (e.g. for 1° skew, 0.020 inch (0.508 mm) thicklamina with 0.002 inch (0.0508 mm) side groove spacing).

Due to the problems of machining stacks of laminae having skewed and/orinclined side grooves, in the practice of such embodiments the sidegrooves are preferably formed in individual laminae with agroove-forming machine. A preferred method for forming grooves on theedge portion of individual laminae, assembling the laminae into a mastertool, and replicating the microstructured surface of the assembledlaminae is described in U.S. Patent Application Publication No.2004/0175541, entitled “Methods of Making Microstructured Lamina andApparatus” filed Mar. 6, 2003, incorporated herein by reference. U.S.Patent Application Publication No. 2004/0175541 was concurrently filedwith Provisional Patent Application Ser. No. 60/452464, to which thepresent application claims priority.

To form a master tool of suitable size for forming retroreflectivesheeting, a plurality of toolings (also referred to as tiles) are formedby electroplating the surface of the master tool to form negativecopies, subsequently electroplating the negative copies to form positivecopies, electroplating the positive copies to form a second generationnegative copies, etc. The positive copy has the same cube corner elementstructure as the master tool, whereas the negative copy is the cubecavity replica. Accordingly, a negative copy tool is employed to make apositive copy (i.e. cube corner element) sheeting whereas, a positivecopy tool is employed to make a negative copy (i.e. cube corner cavity)sheeting. Further, retroreflective sheeting may comprise combination ofcube corner elements and cube corner cavity microstructures.Electroforming techniques such as described in U.S. Pat. Nos. 4,478,769and 5,156,863 (Pricone) as well as U.S. Pat. No. 6,159,407 (Krinke) areknown. Tiling such toolings together can then assemble a master tool ofthe desired size. In the present invention the toolings are typicallytiled in the same orientation.

As used herein, “sheeting” refers to a thin piece of polymeric (e.g.synthetic) material upon which cube corner microstructures have beenformed. The sheeting may be of any width and length, such dimension onlybeing limited by the equipment (e.g. width of the tool, width of theslot die orifice, etc.) from which the sheeting was made. The thicknessof retroreflective sheeting typically ranges from about 0.004 inches(0.1016 mm) to about 0.10 inches (2.54 mm). Preferably the thickness ofretroreflective sheeting is less than about 0.020 inches (0.508 mm) andmore preferably less than about 0.014 inches (0.3556 mm). Theretroreflective sheeting may further include surface layers such as sealfilms or overlays. In the case of retroreflective sheeting, the width istypically at least 30 inches (122 cm) and preferably at least 48 inches(76 cm). The sheeting is typically continuous in its length for up toabout 50 yards (45.5 m) to 100 yards (91 m) such that the sheeting isprovided in a conveniently handled roll-good. Alternatively, however,the sheeting may be manufactured as individual sheets rather than as aroll-good. In such embodiments, the sheets preferably correspond indimensions to the finished article. For example, the retroreflectivesheeting, may have the dimensions of a standard U.S. sign (e.g. 30inches by 30 inches (76 cm by 76 cm) and thus the microstructured toolemployed to prepare the sheeting may have about the same dimensions.Smaller articles such as license plates or reflective buttons may employsheeting having correspondingly smaller dimensions.

The retroreflective sheet is preferably manufactured as an integralmaterial, i.e. wherein the cube-corner elements are interconnected in acontinuous layer throughout the dimension of the mold, the individualelements and connections therebetween comprising the same material. Thesurface of the sheeting opposing the microprismatic surface is typicallysmooth and planar, also being referred to as the “land layer”. Thethickness of the land layer (i.e. the thickness excluding that portionresulting from the replicated microstructure) is between 0.001 and 0.100inches and preferably between 0.003 and 0.010 inches. Manufacture ofsuch sheeting is typically achieved by casting a fluid resin compositiononto the tool and allowing the composition to harden to form a sheet. Apreferred method for casting fluid resin onto the tool is described inU.S. Patent Application Publication No. 2004/0173920, entitled “Methodof Making Retroreflective Sheeting and Slot Die Apparatus”. U.S. PatentApplication Publication No. 2004/0173920 was concurrently filed withProvisional Patent Application Ser. No. 60/452464, to which the presentinvention claims priority.

Optionally, however, the tool can be employed as an embossing tool toform retroreflective articles, such as described in U.S. Pat. No.4,601,861 (Pricone). Alternatively, the retroreflective sheeting can bemanufactured as a layered product by casting the cube-corner elementsagainst a preformed film as taught in PCT application No. WO 95/11464and U.S. Pat. No. 3,684,348, or by laminating a preformed film topreformed cube-corner elements. In doing so the individual cube-cornerelements are interconnected by the preformed film. Further, the elementsand film are typically comprised of different materials.

In the manufacture of the retroreflective sheeting, it is preferred thatthe channels of the tool are roughly aligned with the direction of theadvancing tool as further described in U.S. Patent Application Ser. No.60/452605, entitled “Methods of Making Retroreflective Sheeting andArticles”, filed Mar. 6, 2003. U.S. Patent Application Ser. No.60/452605 was filed concurrently with Provisional Patent ApplicationSer. No. 60/452464, to which the present invention claims priority.Accordingly, prior to any further manufacturing steps, the primarygroove of the sheeting would be substantially parallel to the edge ofthe roll of the sheeting. The present inventors have found thatorienting the channels in this downweb manner allows for fasterreplication than when the primary groove is oriented cross web. It issurmised that the primary groove and other cube structures combine toform channels for improved resin flow.

Suitable resin compositions for the retroreflective sheeting of thisinvention are preferably transparent materials that are dimensionallystable, durable, weatherable, and readily formable into the desiredconfiguration. Examples of suitable materials include acrylics, whichhave an index of refraction of about 1.5, such as Plexiglas brand resinmanufactured by Rohm and Haas Company; polycarbonates, which have anindex of refraction of about 1.59; reactive materials such as thermosetacrylates and epoxy acrylates; polyethylene based ionomers, such asthose marketed under the brand name of SURLYN by E.I. Dupont de Nemoursand Co., Inc.; (poly)ethylene-co-acrylic acid; polyesters;polyurethanes; and cellulose acetate butyrates. Polycarbonates areparticularly suitable because of their toughness and relatively higherrefractive index, which generally contributes to improvedretroreflective performance over a wider range of entrance angles. Thesematerials may also include dyes, colorants, pigments, UV stabilizers, orother additives.

A specular reflective coating such as a metallic coating can be placedon the backside of the cube-corner elements. The metallic coating can beapplied by known techniques such as vapor depositing or chemicallydepositing a metal such as aluminum, silver, or nickel. A primer layermay be applied to the backside of the cube-corner elements to promotethe adherence of the metallic coating. In addition to or in lieu of ametallic coating, a seal film can be applied to the backside of thecube-corner elements; see, for example, U.S. Pat. Nos. 4,025,159 and5,117,304. The seal film maintains an air interface at the backside ofthe cubes that enables total internal reflection at the interface andinhibits the entry of contaminants such as soil and/or moisture. Furthera separate overlay film may be utilized on the viewing surface of thesheeting for improved (e.g. outdoor) durability or to provide an imagereceptive surface. Indicative of such outdoor durability is maintainingsufficient brightness specifications such as called out in ASTMD49560-1a after extended durations of weathering (e.g. 1 year, 3 years).Further the CAP Y whiteness is preferably greater than 30 before andafter weathering.

An adhesive layer also can be disposed behind the cube-corner elementsor the seal film to enable the cube-corner retroreflective sheeting tobe secured to a substrate. Suitable substrates include wood, aluminumsheeting, galvanized steel, polymeric materials such as polymethylmethacrylates, polyesters, polyamids, polyvinyl fluorides,polycarbonates, polyvinyl chlorides, polyurethanes, and a wide varietyof laminates made from these and other materials.

With reference to FIG. 6, the laminae are preferably aligned vertically.In doing so, upon replication a row of elements is derived from eachlamina. Alternatively, however, these same optical features may bederived from horizontally aligned laminae. The common plane that a faceof each element within a row share to within about 3-4 microns may varyslightly (e.g. less than 1°) for horizontally aligned laminae. It isevident that a row of cubes was derived from a lamina due to thepresence of slight vertical or horizontal misalignments as can beobserved with, for example, scanning electron microscopy.

Regardless of the method of making the retroreflective sheeting orwhether the master tool was derived from a lamina technique or othertechnique, the sheeting of the invention has certain unique opticalfeatures that can be detected by viewing the sheeting with a microscopeor interferometer as previously described.

In one aspect, the retroreflective sheeting comprises a row of cubecorner elements or an array of cube corner element wherein the includedangle between a first and second concurrent element in a row differsfrom the included angle between a second and a third concurrent elementin the row. With respect to the sheeting, the row is defined by theelements wherein a face of each element within the row shares a commonplane (e.g. primary groove face, working surface 12 or 14). Themagnitude of the difference in included angle between adjacent cubes aswell as other preferred characteristics (e.g. arranged in a repeatingpattern, common peak height, bisector planes that range form beingmutually nominally parallel to non-parallel by less than 1°) within arow or array is the same as previous described with respect to thelamina.

Alternatively or in combination thereof, the retroreflective sheetingcomprises a row or an array of cube corner elements (e.g. PG cube cornerelements) wherein at least a portion of the elements in a row or arrayare predominantly sideways canted, the elements having an alignmentangles between 45° and 135° and/or having an alignment angle between225° and 315° relative to the dihedral edge that is substantiallyperpendicular to a row of elements in plan view. In preferredembodiments, the retroreflective sheeting comprises a row of cube cornerelements or an array having cube corner elements having each of thesealignment angles. Such array is substantially free of predominantlyforward canted or predominantly backward canted cube corner elements.The retroreflective sheeting comprising predominantly sideways cantedcube corner elements may further comprise any of the characteristicspreviously described with regard to the lamina.

Alternatively or in combination thereof, the retroreflective sheetingcomprises skewed and/or inclined grooves. Hence, the row or the arraywherein at least two adjacent grooves and preferably all the grooves ofthe (e.g. side) groove set are non-parallel by amount ranging fromgreater than nominally parallel to about 1° and may further include thevarious attributes described with regard to lamina comprising thisfeature.

In another aspect, alone or in combination with differing includedangles and/or sideways canting, the retroreflective sheeting maycomprise a row or elements or an array wherein the grooves of the sidegroove set are nominally parallel to each other, yet range from beingnominally parallel to non-parallel to reference plane 28.

The retroreflective sheeting is useful for a variety of uses such astraffic signs, pavement markings, vehicle markings and personal safetyarticles, in view of its high retroreflected brightness. The coefficientof retroreflection, R_(A), may be measured according to US Federal TestMethod Standard 370 at −4° entrance, 0° orientation, at variousobservation angles. The resulting sheeting meets brightnessspecifications called out in ASTM D4956-1a “The Standard Specificationfor Retroreflective Sheeting for Traffic Control” for Type IX sheeting.Additionally, specified brightness minimums are significantly exceededfor −4° entrance, an average of 0° and 90° orientation, 0° presentationand various observation angles. The brightness is preferably at least625 candelas per lux per square meter (CPL), more preferably at least650 CPL, even more preferably at least 675 CPL, and most preferably atleast 700 CPL at an observation angle of 0.2°. Alternatively, andpreferably in addition thereto, the brightness at an observation angleof 0.33° is preferably at least 575 CPL, more preferably at least 600CPL, even more preferably at least 625 CPL, and most preferably at least650 CPL. In addition or in the alternative, the brightness at anobservation angle of 0.5° is preferably at least 375 CPL, morepreferably at least 400 CPL, even more preferably at least 425 CPL, andmost preferably at least 450 CPL. Further, the brightness at anobservation angle of 1.0° is preferably at least 80 CPL, more preferablyat least 100 CPL, and most preferably at least 120 CPL. Likewise, thebrightness at an observation angle of 1.5° is preferably at least 20 CPLand more preferably at least 25 CPL. The retroreflective sheeting maycomprise any combination of brightness criteria just stated.

Improved brightness in the region around 0.5 observation angle (i.e. 0.4to 0.6) is particularly important for viewing traffic signs (e.g. rightshould mounted) from passenger vehicles at distances of roughly 200 to400 feet and for the viewing of traffic signs (e.g. right shouldmounted) from drivers of large trucks at distances of about 450 to 950feet.

Objects and advantages of the invention are further illustrated by thefollowing examples, but the particular materials and amounts thereofrecited in the examples, as well as other conditions and details, shouldnot be construed to unduly limit the invention.

EXAMPLES 1A AND 1B

Grooves were formed in individual lamina, the individual laminaassembled, and the microstructured surface replicated as described inpreviously cited U.S. Patent Application Publication No. 2004/0175541.U.S. Patent Application Publication No. 2004/0175541 was filedconcurrently with Provisional Patent Application Ser. No. 60/452464 towhich the present application claims priority. All the machined laminaehad the geometry depicted in FIGS. 6 and 7, with slight variations dueto varying the half angle error, skew and inclination of the sidegrooves. The lamina thickness was 0.0075 inches (0.1905 mm) and the sidegroove spacing was 0.005625 inches (0.1428 mm) except for the slightvariations just described. A repeating pattern of eight cubes wassequentially formed on each lamina. This repeating pattern of cubes wasformed by varying the half angle errors, skew, and inclination of theside grooves as set forth in forthcoming Tables 10-14. Each row in thetables defines the parameters used during machining of an individualside groove. The cube corner dihedral errors, as defined in FIG. 22, areformed by the two adjacent side grooves that intersect the primarygroove surface to form each cube. Hence, the rows defining dihedralangle errors are offset in the table to clarify their adjacent sidegrooves.

Eight laminae that differed with regard to the angle error and/or skewand/or inclination of the side grooves were formed such that thedihedral angle errors reported in each of the following Tables 10-14were obtained with the exception of Table 13 wherein the skew of aportion of the side grooves was modified.

Lamina 1 and Lamina 2

The side groove parameters of the first lamina as well as the secondlamina, the second lamina being an opposing lamina to the first lamina,are reported in Tables 10 and 11, respectively. The primary groove halfangle error was −8 arc minutes for all the primary grooves. Side groovenominal included angles (the angles required to produce orthogonalcubes) were 75.226° and 104.774°. The included angle error for all sidegrooves was −9.2 arc minutes, resulting in actual side groove includedangles of 75.073° and 104.621°. While the included angle error wasconstant for the side grooves, the half angle errors were varied. Halfangle errors for the first lamina type ranged from −14.8 arc minutes to5.6 arc minutes as shown in column 3 of Table 10. The half angle errorsare presented in groups of two (totaling −9.2 arc minutes) correspondingto the two half angles for each side groove. The dihedral 2-3 errorresults from the combination of half angle errors on adjacent sidegrooves and is summarized in column 4. Dihedral 2-3 errors varied from−1.6 arc minutes to −16.8 arc minutes for the first lamina.

Skew and inclination are set forth in columns five and six of Table 10,respectively. Skew ranged from −8.0 arc minutes to 15.0 arc minutes forthe first lamina. Inclination varied from −6.1 arc minutes to 10.8 arcminutes. The 1-2 and 1-3 dihedral errors resulting from skew andinclination of the side grooves are shown in the final two columns. Notethat dihedral errors 1-2 and 1-3 varied in opposition, with at least onecube in the lamina comprising dihedral errors 1-2 and 1-3 with differentmagnitudes and/or signs.

The side grooves of the second lamina, is summarized in Table 11 and isclosely related to the lamina of Table 10. The first and second columns,that set forth the nominal side groove angle as well as side grooveincluded angle error, are identical. All other columns for side grooveparameters (half angle errors, skew and inclination) as well as dihedralangle errors are inverted in relation to Table 10. This reflects thefact that an opposing lamina is optically identical to its counterpartexcept rotated 180° about the z-axis.

Lamina 4, Lamina 6 and Lamina 8

For simplicity, the side groove parameters of the fourth, sixth, andeight lamina that are respectively opposing the third, fifth and seventhlamina are not reiterated since the side grooves parameter have thissame inverted relationship as just described.

Lamina 3

The side groove parameter of the third lamina is set forth in Table 12.Primary groove half angle error was −8 arc minutes. The basic geometry(dimensions and nominal side groove included angles) was the same as thefirst lamina type. The actual included angle error for all side grooveswas again −9.2 arc minutes. Half angle errors for the second lamina typeside grooves ranged from −14.8 arc minutes to 5.6 arc minutes. Dihedral2-3 errors varied from −1.6 arc minutes to −16.8 arc minutes. Skewranged from −14.0 arc minutes to 21.3 arc minutes while inclinationvaried from −12.7 arc minutes to 16.8 arc minutes for this lamina type.The 1-2 and 1-3 dihedral errors (shown in the final two columns) variedin opposition.

Lamina 5

The groove parameters of the fifth lamina is set forth in Table 13. Theprimary groove half angle error was −4 arc minutes. The basic geometry(dimensions and nominal side groove included angles) was the same as thepreceding laminae. The included angle error for all side grooves was−1.6 arc minutes, resulting in actual side groove included angles of75.199° and 104.747°. Half angle errors for the third lamina type rangedfrom −5.2 arc minutes to 3.6 arc minutes. Dihedral 2-3 errors variedfrom −7.2 arc minutes to 4.0 arc minutes. Skew ranged from −7.0 arcminutes to 9.5 arc minutes while inclination varied from −8.2 arcminutes to 1.4 arc minutes. The 1-2 and 1-3 dihedral errors (shown inthe final two columns) varied in opposition.

Lamina 7

The side groove parameter for the seventh lamina is set forth in Table14. The primary groove half angle error was −4.0 arc minutes. The basicgeometry (dimensions and nominal side groove included angles) was thesame as the first lamina type. The actual included angle error for allside grooves was again −1.6 arc minutes. Half angle errors ranged from−5.2 arc minutes to 3.6 arc minutes. Dihedral 2-3 errors varied from−7.2 arc minutes to 4.0 arc minutes. Skew ranged from −5.3 arc minutesto 5.3 arc minutes while inclination varied from −2.1 arc minutes to 4.6arc minutes for this lamina type. The 1-2 and 1-3 dihedral errors (shownin the final two columns) varied in opposition.

A total of 208 laminae were assembled such that the non-dihedral edgesof the elements of opposing laminae contacted each other to a precisionsuch that the assembly was substantially free of vertical walls (e.g.walls greater than 0.0001 in lateral dimensions). The laminae wereassembled such that the lamina order 1-8 was sequentially repeatedthroughout the assembly and the structured surface of the assembly wasthen replicated by electroforming to create a cube cavity tool. Theassembly and electroforming process is further described in previouslycited U.S. Patent Application Publication No. 2004/0175541. U.S. PatentApplication Publication No. 2004/0175541 was filed concurrently withProvisional Patent Application Ser. No. 60/452464 to which the presentapplication claims priority.

For Example 1A, the tool was used in a compression molding press withthe pressing performed at a temperature of approximately 375° F. (191°C.) to 385° F. (196° C.), a pressure of approximately 1600 psi, and adwell time of 20 seconds. The molded polycarbonate was then cooled toabout 200° F. (100° C.) over 5 minutes.

For Example 2A, molten polycarbonate was cast onto the tool surface asdescribed in previously cited U.S. Patent Application Publication No.2004/0173920. U.S. Patent Application Publication No. 2004/0173920 wasfiled concurrently with Provisional Patent Application Ser. No.60/452464 to which the present application claims priority.

For both Example 1A and 1B, a dual layer seal film comprising 0.7 milspolyester and 0.85 mils amorphous copolyester was applied to thebackside of the cube-corner elements by contacting the amorphouscopolyester containing surface to the microstructured polycarbonate filmsurface in a continuous sealing process. The construction was passedcontinuously through a rubber nip roll having a Teflon sleeve and aheated steel roll. The surface of the rubber nip roll was about 165° F.and the surface of the heated steel roll was about 405° F. The nippressure was about 70 pounds/per linear inch and speed was 20 feet perminute. Brightness retention after sealing was about 70%.

The resulting sheeting meets brightness specifications called out inASTM D4956-1a “The Standard Specification for Retroreflective Sheetingfor Traffic Control” for Type IX sheeting. Additionally, specifiedbrightness minimums are significantly exceeded for −4° entrance, anaverage of 0° and 90° orientation, 0° presentation and variousobservation angles as follows:

TABLE 9 Comparative Comparative Example 1A Example 1B RetroreflectiveRetroreflective Compression Extrusion Sheeting 2 Sheeting 3 MoldedSheeting Sheeting Avg 0/90 Avg 0/90 Avg 0/90 Avg 0/90 Observation AngleCPL CPL CPL CPL 0.2 726 489 788 740 0.33 660 432 748 700 0.5 276 348 554502 1 37 106 141 162 1.5 13 24 32 35

Table 9 shows that the retroreflective sheeting of the present inventionhas a higher brightness at each of the indicated observation angles incomparison to Comparative Retroreflective Sheeting 2 and ComparativeRetroreflective Sheeting 3. The improved brightness in the region around0.5 observation angle is particularly important for viewing trafficsigns (e.g. right should mounted) from passenger vehicles at distancesof roughly 200 to 400 feet and for the viewing of traffic signs (e.g.right should mounted) from drivers of large trucks at distances of about45° to 950 feet.

The sheeting of Example 1A was found to have a measured uniformity indexof 2.04 for total light return within 2.0° observation.

Various modifications and alterations of this invention will becomeapparent to those skilled in the art without departing from the scopeand spirit of this invention.

TABLE 10 Nominal Side Groove Side Groove Side Groove Dihedral DihedralDihedral Included Incl. Angle Half Angle 2-3 Error Skew Inclination 1-3Error 1-2 Error Angle (Deg) Error (min) Errors (min) (min) (min) (min)(min) (min) 75.226 −9.2 −7.2 15.0 2.5 −2.0 −9.2 −16.1 −6.0 104.774 −9.2−7.2 0.0 −0.4 −2.0 −9.2 −6.0 −16.0 75.226 −9.2 −7.2 −7.0 10.8 −2.0 −9.2−7.0 −12.8 104.774 −9.2 −7.2 −8.0 3.1 −2.0 −16.8 −4.8 −5.7 75.226 −9.2−14.8 −7.0 −6.0 5.6 −1.6 3.3 1.9 104.774 −9.2 −7.2 14.7 −1.2 −2.0 −9.2−12.7 −7.0 75.226 −9.2 −7.2 −1.0 2.5 −2.0 −16.8 −5.8 −4.9 104.774 −9.2−14.8 −6.7 −6.1 5.6 −1.6 1.8 3.3 75.226 −9.2 −7.2 15.0 2.5 −2.0

TABLE 11 Nominal Side Groove Side Groove Side Groove Dihedral DihedralDihedral Included Incl. Angle Half Angle 2-3 Error Skew Inclination 1-3Error 1-2 Error Angle (Deg) Error (min) Errors (min) (min) (min) (min)(min) (min) 75.226 −9.2 −2.0 15.0 2.5 −7.2 −1.6 1.8 3.3 104.774 −9.2 5.6−6.7 −6.1 −14.8 −16.8 −5.8 −4.9 75.226 −9.2 −2.0 −1.0 2.5 −7.2 −9.2−12.7 −7.0 104.774 −9.2 −2.0 14.7 −1.2 −7.2 −1.6 3.3 1.9 75.226 −9.2 5.6−7.0 −6.0 −14.8 −16.8 −4.8 −5.7 104.774 −9.2 −2.0 −8.0 3.1 −7.2 −9.2−7.0 −12.8 75.226 −9.2 −2.0 −7.0 10.8 −7.2 −9.2 −6.0 −16.0 104.774 −9.2−2.0 0.0 −0.4 −7.2 −9.2 −16.1 −6.0 75.226 −9.2 −2.0 15.0 2.5 −7.2

TABLE 12 Nominal Side Groove Side Groove Side Groove Dihedral DihedralDihedral Included Incl. Angle Half Angle 2-3 Error Skew Inclination 1-3Error 1-2 Error Angle (Deg) Error (min) Errors (min) (min) (min) (min)(min) (min) 75.226 −9.2 −7.2 21.3 2.0 −2.0 −9.2 −19.8 −8.7 104.774 −9.2−7.2 0.0 3.0 −2.0 −9.2 −8.7 −19.7 75.226 −9.2 −7.2 −7.2 16.8 −2.0 −9.2−10.5 −15.4 104.774 −9.2 −7.2 −14.0 2.6 −2.0 −16.8 −1.4 −1.5 75.226 −9.2−14.8 −6.7 −12.7 5.6 −1.6 7.2 5.0 104.774 −9.2 −7.2 20.5 −1.4 −2.0 −9.2−15.4 −10.6 75.226 −9.2 −7.2 −7.0 2.0 −2.0 −16.8 −1.6 −1.4 104.774 −9.2−14.8 −6.7 −10.5 5.6 −1.6 5.3 7.7 75.226 −9.2 −7.2 21.3 2.0 −2.0

TABLE 13 Nominal Side Groove Side Groove Side Groove Dihedral DihedralDihedral Included Incl. Angle Half Angle 2-3 Error Skew Inclination 1-3Error 1-2 Error Angle (Deg) Error (min) Errors (min) (min) (min) (min)(min) (min) 75.226 −1.6 0.4 2.1 −4.0 −2.0 −1.6 −1.4 3.3 104.774 −1.6 0.40.0 −8.2 −2.0 −1.6 3.3 −1.3 75.226 −1.6 0.4 −4.7 −6.8 −2.0 −1.6 4.7 −1.7104.774 −1.6 0.4 5.1 1.4 −2.0 −7.2 −6.8 −7.6 75.226 −1.6 −5.2 −7.0 1.03.6 4.0 1.5 −1.5 104.774 −1.6 0.4 0.4 −1.8 −2.0 −1.6 −1.9 4.8 75.226−1.6 0.4 9.5 −1.8 −2.0 −7.2 −7.5 −6.8 104.774 −1.6 −5.2 −5.4 1.2 3.6 4.0−1.4 1.4 75.226 −1.6 0.4 2.1 −4.0 −2.0

TABLE 14 Nominal Side Groove Side Groove Side Groove Dihedral DihedralDihedral Included Incl. Angle Half Angle 2-3 Error Skew Inclination 1-3Error 1-2 Error Angle (Deg) Error (min) Errors (min) (min) (min) (min)(min) (min) 75.226 −1.6 0.4 4.7 3.6 −2.0 −1.6 −7.7 −1.5 104.774 −1.6 0.40.0 −2.1 −2.0 −1.6 −1.5 −7.7 75.226 −1.6 0.4 −4.7 3.6 −2.0 −1.6 −1.6−6.8 104.774 −1.6 0.4 0.0 4.6 −2.0 −7.2 −6.8 −7.6 75.226 −1.6 −5.2 −4.73.5 3.6 4.0 −1.6 −1.6 104.774 −1.6 0.4 5.3 1.3 −2.0 −1.6 −6.8 −1.675.226 −1.6 0.4 4.6 3.4 −2.0 −7.2 −7.5 −6.8 104.774 −1.6 −5.2 −5.3 1.33.6 4.0 −1.6 −1.6 75.226 −1.6 0.4 4.7 3.6 −2.0

1. An article comprising preferred geometry cube corner elements wherein at least one cube comprises a 1-2 dihedral angle error and a 1-3 dihedral angle error; wherein the dihedral angle errors vary in opposition.
 2. The article of claim 1 wherein the cube comprises dihedral angle errors having magnitudes between 1 are minute and 60 are minutes.
 3. The article of claim 1 wherein the elements are in a row having dihedral edges that range from being nominally parallel to nonparallel by less than 1°.
 4. The article of claim 1 wherein the dihedral angle errors are varied in a repeating pattern.
 5. An article comprising at least one preferred geometry cube corner element having three dihedral angles errors wherein the dihedral angle errors are different from each other.
 6. The article of claim 5 wherein the dihedral angle errors have magnitudes between 1 are minute and 60 are minutes.
 7. The article of claim 5 comprising a plurality of the elements in a row wherein at least one dihedral edge of the elements range from being nominally parallel to nonparallel by less than 1°.
 8. The article of claim 7 wherein the dihedral angle errors of the elements are varied in a repeating pattern.
 9. The article of claim 7 wherein the article comprises a lamina.
 10. The article of claim 7 wherein the elements each have a first face and the first faces define a primary groove face.
 11. The article of claim 7 wherein the article comprises a tool.
 12. The article of claim 7 wherein the article comprises retroreflective sheeting.
 13. The article of claim 12 wherein the retroreflective sheeting comprises cube corner elements.
 14. The article of claim 12 wherein the retroreflective sheeting comprises cube corner cavities. 